Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(26,30)(27,33)(32,35)(34,39);(1,2,5,3)(4,6,8,7)(12,20)(17,22)(19,21)(23,24)(26,30)(31,37)(32,35)(38,40);(1,2,5,3)(4,7,8,6)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,2,5,3)(4,6,8,7)(9,12,15,20)(10,17,14,22)(11,19,13,21)(16,23,18,24);(1,3,5,2)(4,7,8,6)(9,13,15,11)(10,18,14,16)(12,21,20,19)(17,24,22,23)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095214);